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<b>let </b><font color="Maroon">seq1</font> be   <a href="comseq_1.html#NM1">Complex_Sequence</a>;<br/><b>let </b><font color="Maroon">seq2</font> be   <a href="comseq_1.html#NM1">Complex_Sequence</a>;<br/><b>let </b><font color="Maroon">Ns</font> be  <a href="seqm_3.html#V1">increasing</a> <a href="seqm_3.html#NM1">Seq_of_Nat</a>;<br/>





<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_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/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq1</font> <a href="comseq_1.html#K2">+</a> <font color="Maroon">seq2</font></span>)</span> <a href="comseq_3.html#K5">*</a> <font color="Maroon">Ns</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="comseq_1.html#K2">+</a> <font color="Maroon">seq2</font></span>)</span> <a href="comseq_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">Ns</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="comseq_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">Ns</font> <a href="seqm_3.html#K2">.</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">seq2</font> <a href="comseq_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">Ns</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="comseq_1.html#D4">COMSEQ_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq1</font> <a href="comseq_3.html#K5">*</a> <font color="Maroon">Ns</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">seq2</font> <a href="comseq_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">Ns</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq1</font> <a href="comseq_3.html#K5">*</a> <font color="Maroon">Ns</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">seq2</font> <a href="comseq_3.html#K5">*</a> <font color="Maroon">Ns</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq1</font> <a href="comseq_3.html#K5">*</a> <font color="Maroon">Ns</font></span>)</span> <a href="comseq_1.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">seq2</font> <a href="comseq_3.html#K5">*</a> <font color="Maroon">Ns</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="E2:9"/>
<span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="comseq_1.html#K2">+</a> <font color="Maroon">seq2</font></span>)</span> <a href="comseq_3.html#K5">*</a> <font color="Maroon">Ns</font> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="comseq_3.html#K5">*</a> <font color="Maroon">Ns</font></span>)</span> <a href="comseq_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">seq2</font> <a href="comseq_3.html#K5">*</a> <font color="Maroon">Ns</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="9_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/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq1</font> <a href="comseq_1.html#K6">-</a> <font color="Maroon">seq2</font></span>)</span> <a href="comseq_3.html#K5">*</a> <font color="Maroon">Ns</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="comseq_1.html#K6">-</a> <font color="Maroon">seq2</font></span>)</span> <a href="comseq_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">Ns</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="comseq_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">Ns</font> <a href="seqm_3.html#K2">.</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">seq2</font> <a href="comseq_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">Ns</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq1</font> <a href="comseq_3.html#K5">*</a> <font color="Maroon">Ns</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">seq2</font> <a href="comseq_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">Ns</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq1</font> <a href="comseq_3.html#K5">*</a> <font color="Maroon">Ns</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">seq2</font> <a href="comseq_3.html#K5">*</a> <font color="Maroon">Ns</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq1</font> <a href="comseq_3.html#K5">*</a> <font color="Maroon">Ns</font></span>)</span> <a href="comseq_1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">seq2</font> <a href="comseq_3.html#K5">*</a> <font color="Maroon">Ns</font></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font>
<b>by </b><i/>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:9"/>
<span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="comseq_1.html#K6">-</a> <font color="Maroon">seq2</font></span>)</span> <a href="comseq_3.html#K5">*</a> <font color="Maroon">Ns</font> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="comseq_3.html#K5">*</a> <font color="Maroon">Ns</font></span>)</span> <a href="comseq_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">seq2</font> <a href="comseq_3.html#K5">*</a> <font color="Maroon">Ns</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="9_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/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq1</font> <a href="comseq_1.html#K3">(#)</a> <font color="Maroon">seq2</font></span>)</span> <a href="comseq_3.html#K5">*</a> <font color="Maroon">Ns</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="comseq_1.html#K3">(#)</a> <font color="Maroon">seq2</font></span>)</span> <a href="comseq_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">Ns</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="comseq_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">Ns</font> <a href="seqm_3.html#K2">.</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">seq2</font> <a href="comseq_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">Ns</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="comseq_1.html#D5">COMSEQ_1:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq1</font> <a href="comseq_3.html#K5">*</a> <font color="Maroon">Ns</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">seq2</font> <a href="comseq_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">Ns</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq1</font> <a href="comseq_3.html#K5">*</a> <font color="Maroon">Ns</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">seq2</font> <a href="comseq_3.html#K5">*</a> <font color="Maroon">Ns</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq1</font> <a href="comseq_3.html#K5">*</a> <font color="Maroon">Ns</font></span>)</span> <a href="comseq_1.html#K3">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">seq2</font> <a href="comseq_3.html#K5">*</a> <font color="Maroon">Ns</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="E6:9"/>
<span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="comseq_1.html#K3">(#)</a> <font color="Maroon">seq2</font></span>)</span> <a href="comseq_3.html#K5">*</a> <font color="Maroon">Ns</font> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="comseq_3.html#K5">*</a> <font color="Maroon">Ns</font></span>)</span> <a href="comseq_1.html#K3">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">seq2</font> <a href="comseq_3.html#K5">*</a> <font color="Maroon">Ns</font></span>)</span>
 <b>by </b><i><a class="ref" href="comseq_1.html#T6">COMSEQ_1:6</a></i>;<br/>


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