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<b>let </b><font color="Maroon" title="c2">seq</font>, <font color="Maroon" title="c3">seq1</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"> ( <font color="Maroon" title="c2">seq</font> is    <a href="valued_0.html#M1" title="VALUED_0:mode.1">subsequence</a> of <font color="Maroon" title="c3">seq1</font> implies (  <a href="comseq_3.html#K3" title="COMSEQ_3:func.3">Re</a> <font color="Maroon" title="c2">seq</font> is    <a href="valued_0.html#M1" title="VALUED_0:mode.1">subsequence</a> of  <a href="comseq_3.html#K3" title="COMSEQ_3:func.3">Re</a> <font color="Maroon" title="c3">seq1</font> &amp;  <a href="comseq_3.html#K4" title="COMSEQ_3:func.4">Im</a> <font color="Maroon" title="c2">seq</font> is    <a href="valued_0.html#M1" title="VALUED_0:mode.1">subsequence</a> of  <a href="comseq_3.html#K4" title="COMSEQ_3:func.4">Im</a> <font color="Maroon" title="c3">seq1</font> ) )</span><br/>



<b>assume </b><a NAME="E1:61"/>
<font color="Maroon" title="c2">seq</font> is    <a href="valued_0.html#M1" title="VALUED_0:mode.1">subsequence</a> of <font color="Maroon" title="c3">seq1</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#K3" title="COMSEQ_3:func.3">Re</a> <font color="Maroon" title="c2">seq</font> is    <a href="valued_0.html#M1" title="VALUED_0:mode.1">subsequence</a> of  <a href="comseq_3.html#K3" title="COMSEQ_3:func.3">Re</a> <font color="Maroon" title="c3">seq1</font> &amp;  <a href="comseq_3.html#K4" title="COMSEQ_3:func.4">Im</a> <font color="Maroon" title="c2">seq</font> is    <a href="valued_0.html#M1" title="VALUED_0:mode.1">subsequence</a> of  <a href="comseq_3.html#K4" title="COMSEQ_3:func.4">Im</a> <font color="Maroon" title="c3">seq1</font> )</span><br/>

<b>then consider </b><font color="Maroon" title="c4">Nseq</font> being  <a href="valued_0.html#V7" title="VALUED_0:attr.7">increasing</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> <b> such that </b><br/><a NAME="E3:61"/><i><font color="Green" title="E38">A1</font></i>: 
<font color="Maroon" title="c2">seq</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">seq1</font> <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">*</a> <font color="Maroon" title="c4">Nseq</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="E4:61"/>
 <a href="comseq_3.html#K3" title="COMSEQ_3:func.3">Re</a> <font color="Maroon" title="c2">seq</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K3" title="COMSEQ_3:func.3">Re</a> <font color="Maroon" title="c3">seq1</font></span>)</span> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c4">Nseq</font>
 
<b>by </b><i><a class="txt" href="#E3:61"><i><font color="Green" title="E38">A1</font></i></a>, <a class="ref" href="comseq_3.html#T22" target="_self" title="COMSEQ_3:th.22">Th22</a></i>;<br/>
<b>hence </b><a NAME="E5:61"/>
 <a href="comseq_3.html#K3" title="COMSEQ_3:func.3">Re</a> <font color="Maroon" title="c2">seq</font> is    <a href="valued_0.html#M1" title="VALUED_0:mode.1">subsequence</a> of  <a href="comseq_3.html#K3" title="COMSEQ_3:func.3">Re</a> <font color="Maroon" title="c3">seq1</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> <font color="Maroon" title="c2">seq</font> is    <a href="valued_0.html#M1" title="VALUED_0:mode.1">subsequence</a> of  <a href="comseq_3.html#K4" title="COMSEQ_3:func.4">Im</a> <font color="Maroon" title="c3">seq1</font></span><br/>

<a NAME="E6:61"/>
 <a href="comseq_3.html#K4" title="COMSEQ_3:func.4">Im</a> <font color="Maroon" title="c2">seq</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K4" title="COMSEQ_3:func.4">Im</a> <font color="Maroon" title="c3">seq1</font></span>)</span> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c4">Nseq</font>
 
<b>by </b><i><a class="txt" href="#E3:61"><i><font color="Green" title="E38">A1</font></i></a>, <a class="ref" href="comseq_3.html#T22" target="_self" title="COMSEQ_3:th.22">Th22</a></i>;<br/>
<b>hence </b><a NAME="E7:61"/>
 <a href="comseq_3.html#K4" title="COMSEQ_3:func.4">Im</a> <font color="Maroon" title="c2">seq</font> is    <a href="valued_0.html#M1" title="VALUED_0:mode.1">subsequence</a> of  <a href="comseq_3.html#K4" title="COMSEQ_3:func.4">Im</a> <font color="Maroon" title="c3">seq1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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