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<div class="add">

<b>let </b><font color="Maroon">seq</font> be   <a href="comseq_1.html#NM1">Complex_Sequence</a>;<br/><b>let </b><font color="Maroon">h1</font> be   <a href="partfun1.html#NM1">PartFunc</a> of  <a href="numbers.html#K2">COMPLEX</a> , <a href="numbers.html#K2">COMPLEX</a> ;<br/><b>let </b><font color="Maroon">h2</font> be   <a href="partfun1.html#NM1">PartFunc</a> of  <a href="numbers.html#K2">COMPLEX</a> , <a href="numbers.html#K2">COMPLEX</a> ;<br/>





<b>assume </b><a NAME="E1:17"/><i><font color="Green">E53</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">seq</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">h1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">h2</font></span>)</span>
 ;<br/>

<a NAME="E2:17"/>
( <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">h1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">h2</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h1</font> &amp; <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">h1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">h2</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h2</font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E3:17"/><b>then </b><i><font color="Green">E55</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">seq</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h1</font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">seq</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h2</font> )
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E4:17"/><i><font color="Green">E56</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">seq</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="comseq_1.html#K2">+</a> <font color="Maroon">h2</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="comseq_1.html#D2">COMSEQ_1:def 2</a></i>;<br/>
<a NAME="E5:17"/><i><font color="Green">E69</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">seq</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="comseq_1.html#K6">-</a> <font color="Maroon">h2</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="cfunct_1.html#T4">CFUNCT_1:4</a></i>;<br/>
<a NAME="E6:17"/><i><font color="Green">E70</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">seq</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="comseq_1.html#K3">(#)</a> <font color="Maroon">h2</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="comseq_1.html#D3">COMSEQ_1:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><a NAME="E1:17_1"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">seq</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="comseq_1.html#K2">+</a> <font color="Maroon">h2</font></span>)</span>
 <b>by </b><i>, </i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="comseq_1.html#K2">+</a> <font color="Maroon">h2</font></span>)</span> <a href="cfcont_1.html#K1" target="_self">*</a> <font color="Maroon">seq</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="comseq_1.html#K2">+</a> <font color="Maroon">h2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="complex1.html#K8">+</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="cfunct_1.html#T3">CFUNCT_1:3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="cfcont_1.html#K1" target="_self">*</a> <font color="Maroon">seq</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="complex1.html#K8">+</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="cfcont_1.html#K1" target="_self">*</a> <font color="Maroon">seq</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="complex1.html#K8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h2</font> <a href="cfcont_1.html#K1" target="_self">*</a> <font color="Maroon">seq</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="cfcont_1.html#K1" target="_self">*</a> <font color="Maroon">seq</font></span>)</span> <a href="comseq_1.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">h2</font> <a href="cfcont_1.html#K1" target="_self">*</a> <font color="Maroon">seq</font></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font>
<b>by </b><i><a class="ref" href="comseq_1.html#D4">COMSEQ_1:def 4</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:17"/>
<span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="comseq_1.html#K2">+</a> <font color="Maroon">h2</font></span>)</span> <a href="cfcont_1.html#K1" target="_self">*</a> <font color="Maroon">seq</font> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="cfcont_1.html#K1" target="_self">*</a> <font color="Maroon">seq</font></span>)</span> <a href="comseq_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="cfcont_1.html#K1" target="_self">*</a> <font color="Maroon">seq</font></span>)</span>
 <b>by </b><i><a class="ref" href="comseq_1.html#T6">COMSEQ_1:6</a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><a NAME="E1:17_2"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">seq</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="comseq_1.html#K6">-</a> <font color="Maroon">h2</font></span>)</span>
 <b>by </b><i>, </i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="comseq_1.html#K6">-</a> <font color="Maroon">h2</font></span>)</span> <a href="cfcont_1.html#K1" target="_self">*</a> <font color="Maroon">seq</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="comseq_1.html#K6">-</a> <font color="Maroon">h2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="complex1.html#K11">-</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="cfunct_1.html#T4">CFUNCT_1:4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="cfcont_1.html#K1" target="_self">*</a> <font color="Maroon">seq</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="cfcont_1.html#K1" target="_self">*</a> <font color="Maroon">seq</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h2</font> <a href="cfcont_1.html#K1" target="_self">*</a> <font color="Maroon">seq</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i>, </i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E10:17"/>
<span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="comseq_1.html#K6">-</a> <font color="Maroon">h2</font></span>)</span> <a href="cfcont_1.html#K1" target="_self">*</a> <font color="Maroon">seq</font> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="cfcont_1.html#K1" target="_self">*</a> <font color="Maroon">seq</font></span>)</span> <a href="comseq_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="cfcont_1.html#K1" target="_self">*</a> <font color="Maroon">seq</font></span>)</span>
 <b>by </b><i/>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><a NAME="E1:17_3"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">seq</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="comseq_1.html#K3">(#)</a> <font color="Maroon">h2</font></span>)</span>
 <b>by </b><i>, </i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="comseq_1.html#K3">(#)</a> <font color="Maroon">h2</font></span>)</span> <a href="cfcont_1.html#K1" target="_self">*</a> <font color="Maroon">seq</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="comseq_1.html#K3">(#)</a> <font color="Maroon">h2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="complex1.html#K9">*</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="cfunct_1.html#T5">CFUNCT_1:5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="cfcont_1.html#K1" target="_self">*</a> <font color="Maroon">seq</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="complex1.html#K9">*</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="cfcont_1.html#K1" target="_self">*</a> <font color="Maroon">seq</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="complex1.html#K9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h2</font> <a href="cfcont_1.html#K1" target="_self">*</a> <font color="Maroon">seq</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="cfcont_1.html#K1" target="_self">*</a> <font color="Maroon">seq</font></span>)</span> <a href="comseq_1.html#K3">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">h2</font> <a href="cfcont_1.html#K1" target="_self">*</a> <font color="Maroon">seq</font></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font>
<b>by </b><i><a class="ref" href="comseq_1.html#D5">COMSEQ_1:def 5</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E12:17"/>
<span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="comseq_1.html#K3">(#)</a> <font color="Maroon">h2</font></span>)</span> <a href="cfcont_1.html#K1" target="_self">*</a> <font color="Maroon">seq</font> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="cfcont_1.html#K1" target="_self">*</a> <font color="Maroon">seq</font></span>)</span> <a href="comseq_1.html#K3">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="cfcont_1.html#K1" target="_self">*</a> <font color="Maroon">seq</font></span>)</span>
 <b>by </b><i><a class="ref" href="comseq_1.html#T6">COMSEQ_1:6</a></i>;<br/>


</div>
