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<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="comseq_1.html#NM1">Complex_Sequence</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></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="E1:32"/><i><font color="Green">E21</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></font> <a href="relset_1.html#K8">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:32_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><a NAME="E2:32_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></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">c<sub>3</sub></font> <a href="relset_1.html#K8">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="cfcont_1.html#E1:32"><i><font color="Green">E21</font></i></a></i>;<br/><a NAME="E3:32_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K8">/\</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/><a NAME="E4:32_1"/><b>then </b>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="complex1.html#K5">0c</a> </span><a href="tarski.html#K1">}</a></span></span>)</span> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="cfcont_1.html#E1:32"><i><font color="Green">E21</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E5:32_1"/><b>then </b>
( <font color="Maroon">c<sub>4</sub></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">c<sub>3</sub></font> <a href="cfunct_1.html#K2">^</a> </span>)</span> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="ref" href="cfunct_1.html#D2">CFUNCT_1:def 2</a></i>;<br/><a NAME="E6:32_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="cfunct_1.html#K2">^</a> </span>)</span></span>)</span> <a href="subset_1.html#K8">/\</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E7:32_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="cfunct_1.html#K2">^</a> </span>)</span> <a href="relset_1.html#K8">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/></div><b>end;</b></div>
<a NAME="E3:32"/><b>then </b><i><font color="Green">E22</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="cfunct_1.html#K2">^</a> </span>)</span> <a href="relset_1.html#K8">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E1:32_2"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="cfcont_1.html#T5" target="_self">Th5</a></i>;<br/><a NAME="E2:32_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></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">c<sub>3</sub></font> <a href="relset_1.html#K8">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="cfcont_1.html#E1:32"><i><font color="Green">E21</font></i></a></i>;<br/><a NAME="E3:32_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K8">/\</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/><a NAME="E4:32_2"/><b>then </b>
( <font color="Maroon">c<sub>2</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="complex1.html#K5">0c</a> </span><a href="tarski.html#K1">}</a></span></span>)</span> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="cfcont_1.html#E1:32"><i><font color="Green">E21</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E5:32_2"/><b>then </b><i><font color="Green">E24</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></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">c<sub>3</sub></font> <a href="cfunct_1.html#K2">^</a> </span>)</span> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="ref" href="cfunct_1.html#D2">CFUNCT_1:def 2</a></i>;<br/><a NAME="E6:32_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="cfunct_1.html#K2">^</a> </span>)</span></span>)</span> <a href="subset_1.html#K8">/\</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E7:32_2"/><b>then </b><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="cfunct_1.html#K2">^</a> </span>)</span> <a href="relset_1.html#K8">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="cfunct_1.html#K2">^</a> </span>)</span> <a href="relset_1.html#K8">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="cfcont_1.html#K1" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="cfunct_1.html#K2">^</a> </span>)</span> <a href="relset_1.html#K8">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="cfcont_1.html#E3:32"><i><font color="Green">E22</font></i></a>, <a class="ref" href="cfcont_1.html#T16" target="_self">Th16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="cfunct_1.html#K2">^</a> </span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="cfcont_1.html#E7:32_2"><i><font color="Green">E25</font></i></a>, <a class="ref" href="partfun2.html#T32">PARTFUN2:32</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="complex1.html#K12">"</a> 
<b>by </b><i><a class="txt" href="cfcont_1.html#E5:32_2"><i><font color="Green">E24</font></i></a>, <a class="ref" href="cfunct_1.html#D2">CFUNCT_1:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K8">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="complex1.html#K12">"</a> 
<b>by </b><i><a class="txt" href="cfcont_1.html#E1:32"><i><font color="Green">E21</font></i></a>, <a class="txt" href="cfcont_1.html#E1:32_2"><i><font color="Green">E23</font></i></a>, <a class="ref" href="partfun2.html#T32">PARTFUN2:32</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K8">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="cfcont_1.html#K1" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="complex1.html#K12">"</a> 
<b>by </b><i><a class="txt" href="cfcont_1.html#E1:32"><i><font color="Green">E21</font></i></a>, <a class="ref" href="cfcont_1.html#T16" target="_self">Th16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K8">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="cfcont_1.html#K1" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="comseq_1.html#K7">"</a> </span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="comseq_1.html#D11">COMSEQ_1:def 11</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:32"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="cfunct_1.html#K2">^</a> </span>)</span> <a href="relset_1.html#K8">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="cfcont_1.html#K1" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K8">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="cfcont_1.html#K1" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="comseq_1.html#K7">"</a> 
 <b>by </b><i><a class="ref" href="comseq_1.html#T6">COMSEQ_1:6</a></i>;<br/>


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