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<b>let </b><font color="Maroon">R</font> be   <a href="relat_1.html#NM1">Relation</a>;<br/><b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">b</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">c</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>given </b><font color="Maroon">p</font> being    <a href="rewrite1.html#M1" target="_self">RedSequence</a> of <font color="Maroon">R</font><b> such that </b><a NAME="E1:26"/><i><font color="Green">E21</font></i>: 
( <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="rewrite1.html#D3" target="_self">REWRITE1:def 3</a><br/>

<b>given </b><font color="Maroon">q</font> being    <a href="rewrite1.html#M1" target="_self">RedSequence</a> of <font color="Maroon">R</font><b> such that </b><a NAME="E2:26"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="rewrite1.html#D3" target="_self">REWRITE1:def 3</a><br/>

<a NAME="E3:26"/><i><font color="Green">E31</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 
<b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">r</font> = <font color="Maroon">p</font> <a href="rewrite1.html#K1" target="_self">$^</a> <font color="Maroon">q</font> as    <a href="rewrite1.html#M1" target="_self">RedSequence</a> of <font color="Maroon">R</font> <b>by </b><i><a class="txt" href="rewrite1.html#E1:26"><i><font color="Green">E21</font></i></a>, <a class="txt" href="rewrite1.html#E2"><i><font color="Green">Lemma22</font></i></a>, </i>;<br/>
<b>take </b>
<font color="Maroon">r</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="rewrite1.html#D3" target="_self">REWRITE1:def 3</a><br/>

<b>consider </b><font color="Maroon">p1</font> being   <a href="finseq_1.html#NM1">FinSequence</a>, <font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:26"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T63">FINSEQ_1:63</a></i>;<br/>
<a NAME="E7:26"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">q</font>
 
<b>by </b><i><a class="txt" href="rewrite1.html#E3"><i><font color="Green">Lemma24</font></i></a>, </i>;<br/>
<a NAME="E8:26"/><b>then </b><i><font color="Green">E35</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/>
<a NAME="E9:26"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> 0 or  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 
<b>by </b><i><a class="ref" href="nat_1.html#T19">NAT_1:19</a></i>;<br/>
<a NAME="E10:26"/><b>then </b>
( <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1 )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E11:26"/><b>then </b>
( ( <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9">*&gt;</a></span> ) or 1 <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">p1</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="rewrite1.html#E3"><i><font color="Green">Lemma24</font></i></a>, <a class="txt" href="rewrite1.html#E4"><i><font color="Green">Lemma25</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, <a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a></i>;<br/>
<a NAME="E12:26"/><b>then </b>
( 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p1</font> or (  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> ) )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E13:26"/><b>then </b>
( ( <font color="Maroon">r</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> <a href="funct_1.html#K1">.</a> 1 &amp; <font color="Maroon">p1</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> ) or ( <font color="Maroon">r</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> ) )
 
<b>by </b><i><a class="txt" href="rewrite1.html#E1:26"><i><font color="Green">E21</font></i></a>, <a class="txt" href="rewrite1.html#E2"><i><font color="Green">Lemma22</font></i></a>, <a class="txt" href="rewrite1.html#E3"><i><font color="Green">Lemma24</font></i></a>, <a class="txt" href="rewrite1.html#E4"><i><font color="Green">Lemma25</font></i></a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/>
<b>hence </b><a NAME="E14:26"/>
<font color="Maroon">r</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 ;<br/>

<a NAME="E15:26"/>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font>
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E16:26"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">q</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">q</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E17:26"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">q</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E18:26"/>
<font color="Maroon">r</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font>
 <b>by </b><i><a class="txt" href="rewrite1.html#E2"><i><font color="Green">Lemma22</font></i></a>, <a class="txt" href="rewrite1.html#E4"><i><font color="Green">Lemma25</font></i></a>, <a class="txt" href="rewrite1.html#E5"><i><font color="Green">Lemma26</font></i></a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/>


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