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<b>let </b><font color="Maroon">z1</font> be   <a href="finseq_1.html#NM1">FinSequence</a>, <font color="Maroon">z2</font> be   <a href="finseq_1.html#NM1">FinSequence</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:56_1_2"/><i><font color="Green">E39</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">z1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> &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#K4">dom</a> <font color="Maroon">z1</font> holds <br/><font color="Maroon">z1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a>  <a href="idea_1.html#K15" target="_self">IDEA_P</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">Key</font>,<font color="Olive">i</font></span>)</span>,<font color="Maroon">n</font> ) )
 <b>and </b><br/><a NAME="E2:56_1_2"/><i><font color="Green">E40</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">z2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> &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#K4">dom</a> <font color="Maroon">z2</font> holds <br/><font color="Maroon">z2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a>  <a href="idea_1.html#K15" target="_self">IDEA_P</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">Key</font>,<font color="Olive">i</font></span>)</span>,<font color="Maroon">n</font> ) )
 ;<br/>

<a NAME="E3:56_1_2"/><i><font color="Green">E42</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">z1</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E4:56_1_2"/><i><font color="Green">E43</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">z2</font>
 
<b>by </b><i><a class="txt" href="idea_1.html#E3:56_1_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:56_1_2_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">r</font>
 ;<br/><a NAME="E2:56_1_2_1"/><b>then </b><i><font color="Green">E45</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">z2</font>
 <b>by </b><i><a class="txt" href="idea_1.html#E3:56_1_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E3:56_1_2_1"/>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">z1</font>
 <b>by </b><i>, <a class="txt" href="idea_1.html#E2:56_1_2_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>then </b><font color="Maroon">z1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> = 
 <a href="idea_1.html#K15" target="_self">IDEA_P</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">Key</font>,<font color="Maroon">j</font></span>)</span>,<font color="Maroon">n</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">z2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
<b>by </b><i><a class="txt" href="idea_1.html#E3:56_1_2"><i><font color="Green">E42</font></i></a>, </i>
;<br/><b>hence </b><a NAME="E5:56_1_2_1"/>
<font color="Maroon">z1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:56_1_2"/>
<font color="Maroon">z1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z2</font>
 <b>by </b><i><a class="txt" href="idea_1.html#E4:56_1_2"><i><font color="Green">E43</font></i></a>, , <a class="ref" href="finseq_1.html#T17">FINSEQ_1:17</a></i>;<br/>


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