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

<b>set </b><font color="Maroon">qs'</font> = <font color="Maroon">qs</font>,<font color="Maroon">w</font> <a href="fsm_1.html#K2" target="_self">-admissible</a> ;<br/>
<b>let </b><font color="Maroon">it1</font> be    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">OAlph</font>, <font color="Maroon">it2</font> be    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">OAlph</font>;<br/>

<b>assume </b><a NAME="E1:37_1_2"/><i><font color="Green">E42</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">it1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">w</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">w</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> holds <br/><font color="Maroon">it1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> the <a href="fsm_1.html#U4" target="_self">OFun</a> of <font color="Maroon">sfsm</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">qs</font>,<font color="Maroon">w</font> <a href="fsm_1.html#K2" target="_self">-admissible</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> ) )
 ;<br/>

<b>assume </b><a NAME="E2:37_1_2"/><i><font color="Green">E43</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">it2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">w</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">w</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> holds <br/><font color="Maroon">it2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> the <a href="fsm_1.html#U4" target="_self">OFun</a> of <font color="Maroon">sfsm</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">qs</font>,<font color="Maroon">w</font> <a href="fsm_1.html#K2" target="_self">-admissible</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> ) )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_2_1"><b>now </b></a><div class="add"><b>thus </b><a NAME="E1:37_1_2_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">it1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">it2</font>
 <b>by </b><i>, <a class="ref" href="fsm_1.html#T7" target="_self">Th7</a></i>;<br/><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E2:37_1_2_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">it1</font> )
 ;<br/><a NAME="E3:37_1_2_1"/><b>then </b>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">it1</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E4:37_1_2_1"/><b>then </b>
( <font color="Maroon">it1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> the <a href="fsm_1.html#U4" target="_self">OFun</a> of <font color="Maroon">sfsm</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">qs</font>,<font color="Maroon">w</font> <a href="fsm_1.html#K2" target="_self">-admissible</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> &amp; <font color="Maroon">it2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> the <a href="fsm_1.html#U4" target="_self">OFun</a> of <font color="Maroon">sfsm</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">qs</font>,<font color="Maroon">w</font> <a href="fsm_1.html#K2" target="_self">-admissible</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="fsm_1.html#T7" target="_self">Th7</a></i>;<br/><b>hence </b><a NAME="E5:37_1_2_1"/>
<font color="Maroon">it1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">it2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:37_1_2"/>
<font color="Maroon">it1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">it2</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T18">FINSEQ_1:18</a></i>;<br/>


</div>
