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<span class="kw">let </span><font color="Maroon" title="c1">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="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">w</font> being  <a href="trees_3.html#V5" title="TREES_3:attr.5">FinTree-yielding</a>  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>  st ( for <font color="Olive" title="b2">t</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 <font color="Olive" title="b2">t</font> <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="Olive" title="b1">w</font> holds <br/> <a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <font color="Olive" title="b2">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> ) holds <br/> <a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <span class="p1">(<span class="default"><a href="trees_3.html#K13" title="TREES_3:func.13">tree</a> <font color="Olive" title="b1">w</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span><br/><span class="kw">let </span><font color="Maroon" title="c2">w</font> be   <a href="trees_3.html#V5" title="TREES_3:attr.5">FinTree-yielding</a>  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( for <font color="Olive" title="b1">t</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 <font color="Olive" title="b1">t</font> <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="c2">w</font> holds <br/> <a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <font color="Olive" title="b1">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> ) implies  <a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <span class="p1">(<span class="default"><a href="trees_3.html#K13" title="TREES_3:func.13">tree</a> <font color="Maroon" title="c2">w</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )</span><br/>



<span class="kw">assume </span><a NAME="E1:148"/><span class="lab"><font color="Green" title="E56">A1</font></span>: 
for <font color="Olive" title="b1">t</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 <font color="Olive" title="b1">t</font> <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="c2">w</font> holds <br/> <a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <font color="Olive" title="b1">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <span class="p1">(<span class="default"><a href="trees_3.html#K13" title="TREES_3:func.13">tree</a> <font color="Maroon" title="c2">w</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c3">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> <span class="kw"> such that </span><br/><a NAME="E3:148"/><span class="lab"><font color="Green" title="E57">A2</font></span>: 
<font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_3.html#K13" title="TREES_3:func.13">tree</a> <font color="Maroon" title="c2">w</font>
 <span class="kw">and </span><br/><a NAME="E4:148"/><span class="lab"><font color="Green" title="E58">A3</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <span class="p1">(<span class="default"><a href="trees_3.html#K13" title="TREES_3:func.13">tree</a> <font color="Maroon" title="c2">w</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="trees_1.html#D15" title="TREES_1:def.15">TREES_1:def 15</a></span>;<br/>
<a NAME="E5:148"/><span class="lab"><font color="Green" title="E59">A4</font></span>: 
( ( <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</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="c2">w</font> &amp; <font color="Olive" title="b2">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">w</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="c3">p</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="#E3:148"><span class="lab"><font color="Green" title="E57">A2</font></span></a>, <a class="ref" href="trees_3.html#D15" target="_self" title="TREES_3:def.15">Def15</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="148_1"><span class="kw">now </span></a><div class="add"><span class="kw">given </span><font color="Maroon" title="c4">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> , <font color="Maroon" title="c5">q</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><span class="kw"> such that </span><a NAME="E2:148_1"/><span class="lab"><font color="Green" title="E60">A5</font></span>: 
<font color="Maroon" title="c4">k</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="c2">w</font>
 <span class="kw">and </span><br/><a NAME="E3:148_1"/><span class="lab"><font color="Green" title="E61">A6</font></span>: 
<font color="Maroon" title="c5">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">w</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:148_1"/><span class="lab"><font color="Green" title="E62">A7</font></span>: 
<font color="Maroon" title="c3">p</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="c4">k</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="c5">q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <span class="p1">(<span class="default"><a href="trees_3.html#K13" title="TREES_3:func.13">tree</a> <font color="Maroon" title="c2">w</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span><br/><a NAME="E5:148_1"/><span class="lab"><font color="Green" title="E63">A8</font></span>: 
<font color="Maroon" title="c2">w</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</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="c2">w</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:148_1"><span class="lab"><font color="Green" title="E60">A5</font></span></a>, <a class="txt" href="trees_3.html#E61"><span class="lab"><font color="Green" title="E40">Lm4</font></span></a></span>;<br/><a NAME="E6:148_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">w</font> is  <a href="trees_3.html#V2" title="TREES_3:attr.2">constituted-FinTrees</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="trees_3.html#D10" target="_self" title="TREES_3:def.10">Def10</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">t</font> = <font color="Maroon" title="c2">w</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> as   <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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:148_1"><span class="lab"><font color="Green" title="E63">A8</font></span></a>, <a class="ref" href="trees_3.html#D4" target="_self" title="TREES_3:def.4">Def4</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c7">q</font> = <font color="Maroon" title="c5">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="#E4:148_1"><span class="lab"><font color="Green" title="E62">A7</font></span></a>, <a class="ref" href="finseq_1.html#T50" title="FINSEQ_1:th.50">FINSEQ_1:50</a></span>;<br/><a NAME="E9:148_1"/><span class="lab"><font color="Green" title="E64">A9</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">q</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <font color="Maroon" title="c6">t</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:148_1"><span class="lab"><font color="Green" title="E61">A6</font></span></a>, <a class="ref" href="trees_1.html#D15" title="TREES_1:def.15">TREES_1:def 15</a></span>;<br/><a NAME="E10:148_1"/><span class="lab"><font color="Green" title="E65">A10</font></span>: 
 <a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <font color="Maroon" title="c6">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:148"><span class="lab"><font color="Green" title="E56">A1</font></span></a>, <a class="txt" href="#E2:148_1"><span class="lab"><font color="Green" title="E60">A5</font></span></a>, <a class="txt" href="trees_3.html#E61"><span class="lab"><font color="Green" title="E40">Lm4</font></span></a></span>;<br/><a NAME="E11:148_1"/><span class="lab"><font color="Green" title="E66">A11</font></span>: 
 <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="c4">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
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></span>;<br/><a NAME="E12:148_1"/><span class="lab"><font color="Green" title="E67">A12</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">q</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:148_1"><span class="lab"><font color="Green" title="E64">A9</font></span></a>, <a class="txt" href="#E10:148_1"><span class="lab"><font color="Green" title="E65">A10</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E13:148_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">q</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:148_1"><span class="lab"><font color="Green" title="E62">A7</font></span></a>, <a class="txt" href="#E11:148_1"><span class="lab"><font color="Green" title="E66">A11</font></span></a>, <a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></span>;<br/><span class="kw">hence </span><a NAME="E14:148_1"/>
 <a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <span class="p1">(<span class="default"><a href="trees_3.html#K13" title="TREES_3:func.13">tree</a> <font color="Maroon" title="c2">w</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:148"><span class="lab"><font color="Green" title="E58">A3</font></span></a>, <a class="txt" href="#E12:148_1"><span class="lab"><font color="Green" title="E67">A12</font></span></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E7:148"/>
 <a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <span class="p1">(<span class="default"><a href="trees_3.html#K13" title="TREES_3:func.13">tree</a> <font color="Maroon" title="c2">w</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:148"><span class="lab"><font color="Green" title="E58">A3</font></span></a>, <a class="txt" href="#E5:148"><span class="lab"><font color="Green" title="E59">A4</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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