<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">seq</font> be   <a href="comseq_1.html#NM1" title="COMSEQ_1:NM.1">Complex_Sequence</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">h</font> being  <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/> for <font color="Olive" title="b2">Ns</font> being <a href="valued_0.html#V7" title="VALUED_0:attr.7">V32</a> <a href="nat_1.html#NM2" title="NAT_1:NM.2">sequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c1">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="Olive" title="b1">h</font> holds <br/> <a href="comseq_3.html#K4" title="COMSEQ_3:func.4">Im</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">h</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c1">seq</font></span>)</span> <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">*</a> <font color="Olive" title="b2">Ns</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="comseq_3.html#K4" title="COMSEQ_3:func.4">Im</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">h</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">*</a> <font color="Olive" title="b2">Ns</font></span>)</span></span>)</span></span><br/><b>let </b><font color="Maroon" title="c2">h</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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">Ns</font> being <a href="valued_0.html#V7" title="VALUED_0:attr.7">V32</a> <a href="nat_1.html#NM2" title="NAT_1:NM.2">sequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c1">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="c2">h</font> holds <br/> <a href="comseq_3.html#K4" title="COMSEQ_3:func.4">Im</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">h</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c1">seq</font></span>)</span> <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">*</a> <font color="Olive" title="b1">Ns</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="comseq_3.html#K4" title="COMSEQ_3:func.4">Im</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">h</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">*</a> <font color="Olive" title="b1">Ns</font></span>)</span></span>)</span></span><br/><b>let </b><font color="Maroon" title="c3">Ns</font> be  <a href="valued_0.html#V7" title="VALUED_0:attr.7">V32</a> <a href="nat_1.html#NM2" title="NAT_1:NM.2">sequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c1">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="c2">h</font> implies  <a href="comseq_3.html#K4" title="COMSEQ_3:func.4">Im</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">h</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c1">seq</font></span>)</span> <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">*</a> <font color="Maroon" title="c3">Ns</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="comseq_3.html#K4" title="COMSEQ_3:func.4">Im</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">h</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">*</a> <font color="Maroon" title="c3">Ns</font></span>)</span></span>)</span> )</span><br/>





<b>assume </b><a NAME="E1:14"/><i><font color="Green" title="E10">A1</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c1">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="c2">h</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="comseq_3.html#K4" title="COMSEQ_3:func.4">Im</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">h</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c1">seq</font></span>)</span> <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">*</a> <font color="Maroon" title="c3">Ns</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="comseq_3.html#K4" title="COMSEQ_3:func.4">Im</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">h</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">*</a> <font color="Maroon" title="c3">Ns</font></span>)</span></span>)</span></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="comseq_3.html#K4" title="COMSEQ_3:func.4">Im</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">h</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c1">seq</font></span>)</span> <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">*</a> <font color="Maroon" title="c3">Ns</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K4" title="COMSEQ_3:func.4">Im</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">h</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">*</a> <font color="Maroon" title="c3">Ns</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">n</font></span><br/><a NAME="E1:14_1"/>
<font color="Maroon" title="c1">seq</font> <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">*</a> <font color="Maroon" title="c3">Ns</font> is    <a href="valued_0.html#M1" title="VALUED_0:mode.1">subsequence</a> of <font color="Maroon" title="c1">seq</font>
 <b>by </b><i><a class="ref" href="valued_0.html#D17" title="VALUED_0:def.17">VALUED_0:def 17</a></i>;<br/><a NAME="E2:14_1"/><b>then </b><i><font color="Green" title="E11">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="c1">seq</font> <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">*</a> <font color="Maroon" title="c3">Ns</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="c1">seq</font>
 <b>by </b><i><a class="ref" href="valued_0.html#T21" title="VALUED_0:th.21">VALUED_0:21</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="comseq_3.html#K4" title="COMSEQ_3:func.4">Im</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">h</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c1">seq</font></span>)</span> <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">*</a> <font color="Maroon" title="c3">Ns</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">n</font> = 
 <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">h</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c1">seq</font></span>)</span> <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">*</a> <font color="Maroon" title="c3">Ns</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">n</font></span>)</span>
<b>by </b><i><a class="ref" href="comseq_3.html#D4" title="COMSEQ_3:def.4">COMSEQ_3:def 4</a></i>
<br/>.= 
 <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">h</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c1">seq</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">Ns</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>
<br/>.= 
 <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">Ns</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E1:14"><i><font color="Green" title="E10">A1</font></i></a>, <a class="ref" href="funct_2.html#T186" title="FUNCT_2:th.186">FUNCT_2:186</a></i>
<br/>.= 
 <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">*</a> <font color="Maroon" title="c3">Ns</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>
<br/>.= 
 <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">h</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">*</a> <font color="Maroon" title="c3">Ns</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">n</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:14"><i><font color="Green" title="E10">A1</font></i></a>, <a class="txt" href="#E2:14_1"><i><font color="Green" title="E11">A2</font></i></a>, <a class="ref" href="funct_2.html#T186" title="FUNCT_2:th.186">FUNCT_2:186</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="comseq_3.html#K4" title="COMSEQ_3:func.4">Im</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">h</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">*</a> <font color="Maroon" title="c3">Ns</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">n</font>
<b>by </b><i><a class="ref" href="comseq_3.html#D4" title="COMSEQ_3:def.4">COMSEQ_3:def 4</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:14"/>
 <a href="comseq_3.html#K4" title="COMSEQ_3:func.4">Im</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">h</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c1">seq</font></span>)</span> <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">*</a> <font color="Maroon" title="c3">Ns</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="comseq_3.html#K4" title="COMSEQ_3:func.4">Im</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">h</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">*</a> <font color="Maroon" title="c3">Ns</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
