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<b>let </b><font color="Maroon">z1</font> be   <a href="algseq_1.html#NM1">AlgSequence</a> of <font color="Maroon">L</font>, <font color="Maroon">z2</font> be   <a href="algseq_1.html#NM1">AlgSequence</a> of <font color="Maroon">L</font>;<br/>

<b>assume </b><a NAME="E1:47_1_2"/><i><font color="Green">E64</font></i>: 
( ( for <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">M</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> <font color="Maroon">M</font> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Olive">i</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>,1 ) &amp; ( for <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">M</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="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font> ) )
 ;<br/>

<b>assume </b><a NAME="E2:47_1_2"/><i><font color="Green">E65</font></i>: 
( ( for <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">M</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> <font color="Maroon">M</font> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Olive">i</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>,1 ) &amp; ( for <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">M</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="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font> ) )
 ;<br/>

<i><font color="Green">E66</font></i>:  <a href="relat_1.html#K1">dom</a> <font color="Maroon">z1</font> = 
 <a href="numbers.html#K5">NAT</a> 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">z2</font>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">u</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:47_1_2_2"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">z1</font>
 ;<br/><b>then reconsider </b><font color="Maroon">u'</font> = <font color="Maroon">u</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:47_1_2_2_1"/>
( <font color="Maroon">u'</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">M</font> or  <a href="finseq_1.html#K3">len</a> <font color="Maroon">M</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">u'</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:47_1_2_2_1_1"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">u'</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">M</font>
 ;<br/><div class="add"><b>hence </b><font color="Maroon">z1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font> = 
<font color="Maroon">M</font> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">u'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,1
<b>by </b><i><a class="ref" href="polynom8.html#T2" target="_self">Th2</a></i>
<br/>.= 
<font color="Maroon">z2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font>
<b>by </b><i><a class="ref" href="polynom8.html#T3" target="_self">Th3</a>, </i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:47_1_2_2_1_2"/><i><font color="Green">E68</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">M</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">u'</font>
 ;<br/><div class="add"><b>hence </b><font color="Maroon">z1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font> = 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
<b>by </b><i><a class="ref" href="polynom8.html#T2" target="_self">Th2</a></i>
<br/>.= 
<font color="Maroon">z2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font>
<b>by </b><i>, <a class="ref" href="polynom8.html#T3" target="_self">Th3</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:47_1_2"/>
<font color="Maroon">z1</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">z2</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


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