<?xml version="1.0"?>
<div class="add">

<b>thus </b><a NAME="E1:35_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a> </span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> 7 implies  ex <font color="Olive">r</font> being  <a href="nat_1.html#NM1">Nat</a> ex <font color="Olive">T</font>, <font color="Olive">X</font>, <font color="Olive">Y</font> being   <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">s</font> ex <font color="Olive">y</font>, <font color="Olive">z</font> being  <a href="prelamb.html#NM1">type</a> of <font color="Maroon">s</font> st <br/>( <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a> </span>)</span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="prelamb.html#K6" target="_self">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive">X</font> <a href="finseq_1.html#K8">^</a> <font color="Olive">T</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Olive">Y</font></span>)</span>,<font color="Olive">z</font></span><a href="prelamb.html#K6" target="_self">]</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="prelamb.html#K6" target="_self">[</a><span class="default"><font color="Olive">T</font>,<font color="Olive">y</font></span><a href="prelamb.html#K6" target="_self">]</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="prelamb.html#K6" target="_self">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive">X</font> <a href="finseq_1.html#K8">^</a> <span class="p4"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Olive">y</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Olive">Y</font></span>)</span>,<font color="Olive">z</font></span><a href="prelamb.html#K6" target="_self">]</a></span> &amp; <font color="Olive">r</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="prelamb.html#K7" target="_self">size_w.r.t.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="prelamb.html#K7" target="_self">size_w.r.t.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">z</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="prelamb.html#K7" target="_self">size_w.r.t.</a> <font color="Maroon">s</font></span>)</span> <a href="prelamb.html#K8" target="_self">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Olive">X</font> <a href="finseq_1.html#K8">^</a> <font color="Olive">T</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Olive">Y</font></span>)</span></span>)</span></span>)</span> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:35_1_1_1"/><i><font color="Green">E36</font></i>: 
<a href="xboole_0.html#K1">{}</a>  <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a></i>;<br/>
<a NAME="E2:35_1_1_1"/><i><font color="Green">E37</font></i>: 
<a href="xboole_0.html#K1">{}</a>  <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a></i>;<br/>
<b>reconsider </b><font color="Maroon">v</font> =  <a href="xboole_0.html#K1">{}</a>  as    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font> <b>by </b><i><a class="ref" href="trees_1.html#T47">TREES_1:47</a></i>;<br/>
<b>assume </b><a NAME="E4:35_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a> </span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> 7
 ;<br/>

<b>then consider </b><font color="Maroon">w</font> being    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font>, <font color="Maroon">u</font> being    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font>, <font color="Maroon">T</font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">s</font>, <font color="Maroon">X</font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">s</font>, <font color="Maroon">Y</font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">s</font>, <font color="Maroon">y</font> being   <a href="prelamb.html#NM1">type</a> of <font color="Maroon">s</font>, <font color="Maroon">z</font> being   <a href="prelamb.html#NM1">type</a> of <font color="Maroon">s</font><b> such that </b><br/><a NAME="E6:35_1_1_1"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12">*&gt;</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="trees_2.html#K3">.</a> <font color="Maroon">v</font></span>)</span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="prelamb.html#K6" target="_self">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">X</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">T</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Y</font></span>)</span>,<font color="Maroon">z</font></span><a href="prelamb.html#K6" target="_self">]</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="trees_2.html#K3">.</a> <font color="Maroon">w</font></span>)</span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="prelamb.html#K6" target="_self">[</a><span class="default"><font color="Maroon">T</font>,<font color="Maroon">y</font></span><a href="prelamb.html#K6" target="_self">]</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="trees_2.html#K3">.</a> <font color="Maroon">u</font></span>)</span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="prelamb.html#K6" target="_self">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">X</font> <a href="finseq_1.html#K8">^</a> <span class="p4"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">y</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Y</font></span>)</span>,<font color="Maroon">z</font></span><a href="prelamb.html#K6" target="_self">]</a></span> )
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">a</font> =  <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="prelamb.html#K7" target="_self">size_w.r.t.</a> <font color="Maroon">s</font></span>)</span> <a href="prelamb.html#K8" target="_self">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">X</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">T</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Y</font></span>)</span></span>)</span> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i/>;<br/>
<b>take </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="prelamb.html#K7" target="_self">size_w.r.t.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="prelamb.html#K7" target="_self">size_w.r.t.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">a</font>
;<br/>

<b>thus </b><a NAME="E8:35_1_1_1"/>
 ex <font color="Olive">T</font>, <font color="Olive">X</font>, <font color="Olive">Y</font> being   <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">s</font> ex <font color="Olive">y</font>, <font color="Olive">z</font> being  <a href="prelamb.html#NM1">type</a> of <font color="Maroon">s</font> st <br/>( <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a> </span>)</span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="prelamb.html#K6" target="_self">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive">X</font> <a href="finseq_1.html#K8">^</a> <font color="Olive">T</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Olive">Y</font></span>)</span>,<font color="Olive">z</font></span><a href="prelamb.html#K6" target="_self">]</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="prelamb.html#K6" target="_self">[</a><span class="default"><font color="Olive">T</font>,<font color="Olive">y</font></span><a href="prelamb.html#K6" target="_self">]</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="prelamb.html#K6" target="_self">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive">X</font> <a href="finseq_1.html#K8">^</a> <span class="p4"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Olive">y</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Olive">Y</font></span>)</span>,<font color="Olive">z</font></span><a href="prelamb.html#K6" target="_self">]</a></span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="prelamb.html#K7" target="_self">size_w.r.t.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="prelamb.html#K7" target="_self">size_w.r.t.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="prelamb.html#K7" target="_self">size_w.r.t.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="prelamb.html#K7" target="_self">size_w.r.t.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">z</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="prelamb.html#K7" target="_self">size_w.r.t.</a> <font color="Maroon">s</font></span>)</span> <a href="prelamb.html#K8" target="_self">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Olive">X</font> <a href="finseq_1.html#K8">^</a> <font color="Olive">T</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Olive">Y</font></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i>, , </i>;<br/>


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

<b>thus </b><a NAME="E2:35_1_1"/>
( not <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a> </span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> 7 implies  ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/>


</div>
