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<b>let </b><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c2">seq</font> be   <a href="comseq_1.html#NM1" title="COMSEQ_1:NM.1">Complex_Sequence</a>;<br/><b>let </b><font color="Maroon" title="c3">f</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> , <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> ;<br/>





<b>assume </b><a NAME="E1:16"/><i><font color="Green" title="E13">A1</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">seq</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font>
 ;<br/>

<a NAME="E2:16"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="comseq_3.html#K6" title="COMSEQ_3:func.6">^\</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">seq</font>
 
<b>by </b><i><a class="ref" href="cfcont_1.html#T3" target="_self" title="CFCONT_1:th.3">Th3</a></i>;<br/>
<a NAME="E3:16"/><b>then </b><i><font color="Green" title="E14">A2</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="comseq_3.html#K6" title="COMSEQ_3:func.6">^\</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font>
 
<b>by </b><i><a class="txt" href="cfcont_1.html#E1:16"><i><font color="Green" title="E13">A1</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">m</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="cfcont_1.html#K1" title="CFCONT_1:func.1">*</a> <font color="Maroon" title="c2">seq</font></span>)</span> <a href="comseq_3.html#K6" title="COMSEQ_3:func.6">^\</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c4">m</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="cfcont_1.html#K1" title="CFCONT_1:func.1">*</a> <font color="Maroon" title="c2">seq</font></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c1">n</font></span>)</span>
<b>by </b><i><a class="ref" href="comseq_3.html#D6" title="COMSEQ_3:def.6">COMSEQ_3:def 6</a></i>
<br/>.= 
<font color="Maroon" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="cfcont_1.html#E1:16"><i><font color="Green" title="E13">A1</font></i></a>, <a class="ref" href="cfcont_1.html#T16" target="_self" title="CFCONT_1:th.16">Th16</a></i>
<br/>.= 
<font color="Maroon" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="comseq_3.html#K6" title="COMSEQ_3:func.6">^\</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c4">m</font></span>)</span>
<b>by </b><i><a class="ref" href="comseq_3.html#D6" title="COMSEQ_3:def.6">COMSEQ_3:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="cfcont_1.html#K1" title="CFCONT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="comseq_3.html#K6" title="COMSEQ_3:func.6">^\</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c4">m</font>
<b>by </b><i><a class="txt" href="cfcont_1.html#E3:16"><i><font color="Green" title="E14">A2</font></i></a>, <a class="ref" href="cfcont_1.html#T16" target="_self" title="CFCONT_1:th.16">Th16</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:16"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="cfcont_1.html#K1" title="CFCONT_1:func.1">*</a> <font color="Maroon" title="c2">seq</font></span>)</span> <a href="comseq_3.html#K6" title="COMSEQ_3:func.6">^\</a> <font color="Maroon" title="c1">n</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c3">f</font> <a href="cfcont_1.html#K1" title="CFCONT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="comseq_3.html#K6" title="COMSEQ_3:func.6">^\</a> <font color="Maroon" title="c1">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="comseq_1.html#T6" title="COMSEQ_1:th.6">COMSEQ_1:6</a></i>;<br/>


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