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

<b>let </b><font color="Maroon">A</font> be   <a href="chain_1.html#NM3">Cell</a> of <font color="Maroon">d</font>,<font color="Maroon">G</font>, <font color="Maroon">A'</font> be   <a href="chain_1.html#NM3">Cell</a> of <font color="Maroon">d</font>,<font color="Maroon">G</font>;<br/>

<b>given </b><font color="Maroon">l</font> being    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">d</font>, <font color="Maroon">r</font> being    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">d</font><b> such that </b><a NAME="E1:71_1_2"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="chain_1.html#K4" target="_self">cell</a> <font color="Maroon">l</font>,<font color="Maroon">r</font> &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font> holds <br/> ( <font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Maroon">G</font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">i</font> ) ) )
 ;<br/>

<b>given </b><font color="Maroon">l'</font> being    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">d</font>, <font color="Maroon">r'</font> being    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">d</font><b> such that </b><a NAME="E2:71_1_2"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">A'</font> <a href="hidden.html#R1">=</a>  <a href="chain_1.html#K4" target="_self">cell</a> <font color="Maroon">l'</font>,<font color="Maroon">r'</font> &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font> holds <br/> ( <font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Maroon">G</font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">i</font> ) ) )
 ;<br/>

<b>reconsider </b><font color="Maroon">l</font> = <font color="Maroon">l</font>, <font color="Maroon">r</font> = <font color="Maroon">r</font>, <font color="Maroon">l'</font> = <font color="Maroon">l'</font>, <font color="Maroon">r'</font> = <font color="Maroon">r'</font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font>, <a href="numbers.html#K1">REAL</a>  <b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font>;<br/><a NAME="E1:71_1_2_1"/>
( <font color="Maroon">r</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">l</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Maroon">G</font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">r'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">l'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">l'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Maroon">G</font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Maroon">i</font> )
 <b>by </b><i>, <a class="ref" href="chain_1.html#D3" target="_self">Def3</a></i>;<br/><b>hence </b><a NAME="E2:71_1_2_1"/>
( <font color="Maroon">l</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">r</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E5:71_1_2"/><b>then </b>
( <font color="Maroon">l</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">l'</font> &amp; <font color="Maroon">r</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">r'</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>
<b>hence </b><a NAME="E6:71_1_2"/>
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A'</font>
 <b>by </b><i>, <a class="ref" href="chain_1.html#D3" target="_self">Def3</a></i>;<br/>


</div>
