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<b>let </b><font color="Maroon" title="c1">seq1</font>, <font color="Maroon" title="c2">seq2</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">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>  holds <br/> ( <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c2">seq2</font></span>)</span> <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">*</a> <font color="Olive" title="b1">Ns</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">*</a> <font color="Olive" title="b1">Ns</font></span>)</span> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq2</font> <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">*</a> <font color="Olive" title="b1">Ns</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <font color="Maroon" title="c2">seq2</font></span>)</span> <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">*</a> <font color="Olive" title="b1">Ns</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">*</a> <font color="Olive" title="b1">Ns</font></span>)</span> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq2</font> <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">*</a> <font color="Olive" title="b1">Ns</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="valued_1.html#K19" title="VALUED_1:func.19">(#)</a> <font color="Maroon" title="c2">seq2</font></span>)</span> <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">*</a> <font color="Olive" title="b1">Ns</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">*</a> <font color="Olive" title="b1">Ns</font></span>)</span> <a href="valued_1.html#K19" title="VALUED_1:func.19">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq2</font> <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">*</a> <font color="Olive" title="b1">Ns</font></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"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c2">seq2</font></span>)</span> <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">*</a> <font color="Maroon" title="c3">Ns</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">*</a> <font color="Maroon" title="c3">Ns</font></span>)</span> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq2</font> <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">*</a> <font color="Maroon" title="c3">Ns</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <font color="Maroon" title="c2">seq2</font></span>)</span> <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">*</a> <font color="Maroon" title="c3">Ns</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">*</a> <font color="Maroon" title="c3">Ns</font></span>)</span> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq2</font> <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">*</a> <font color="Maroon" title="c3">Ns</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="valued_1.html#K19" title="VALUED_1:func.19">(#)</a> <font color="Maroon" title="c2">seq2</font></span>)</span> <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">*</a> <font color="Maroon" title="c3">Ns</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">*</a> <font color="Maroon" title="c3">Ns</font></span>)</span> <a href="valued_1.html#K19" title="VALUED_1:func.19">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq2</font> <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">*</a> <font color="Maroon" title="c3">Ns</font></span>)</span> )</span><br/>





<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_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"><span class="p2">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c2">seq2</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">*</a> <font color="Maroon" title="c3">Ns</font></span>)</span> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">seq2</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><br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c2">seq2</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 class="p1">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c2">seq2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<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>
<b>by </b><i><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">seq1</font> <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> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq2</font> <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="valued_1.html#T1" title="VALUED_1:th.1">VALUED_1:1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">seq1</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> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq2</font> <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/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">seq1</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> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">seq2</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>
<b>by </b><i><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">*</a> <font color="Maroon" title="c3">Ns</font></span>)</span> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">seq2</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>
<b>by </b><i><a class="ref" href="valued_1.html#T1" title="VALUED_1:th.1">VALUED_1:1</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="E2:3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c2">seq2</font></span>)</span> <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">*</a> <font color="Maroon" title="c3">Ns</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">*</a> <font color="Maroon" title="c3">Ns</font></span>)</span> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq2</font> <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">*</a> <font color="Maroon" title="c3">Ns</font></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"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <font color="Maroon" title="c2">seq2</font></span>)</span> <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">*</a> <font color="Maroon" title="c3">Ns</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">*</a> <font color="Maroon" title="c3">Ns</font></span>)</span> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq2</font> <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">*</a> <font color="Maroon" title="c3">Ns</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="valued_1.html#K19" title="VALUED_1:func.19">(#)</a> <font color="Maroon" title="c2">seq2</font></span>)</span> <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">*</a> <font color="Maroon" title="c3">Ns</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">*</a> <font color="Maroon" title="c3">Ns</font></span>)</span> <a href="valued_1.html#K19" title="VALUED_1:func.19">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq2</font> <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">*</a> <font color="Maroon" title="c3">Ns</font></span>)</span> )</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_2"><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"><span class="p2">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <font color="Maroon" title="c2">seq2</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">*</a> <font color="Maroon" title="c3">Ns</font></span>)</span> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">seq2</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><br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <font color="Maroon" title="c2">seq2</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 class="p1">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <font color="Maroon" title="c2">seq2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<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>
<b>by </b><i><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">seq1</font> <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> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq2</font> <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="cfcont_1.html#T2" target="_self" title="CFCONT_1:th.2">Th2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">seq1</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> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq2</font> <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/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">seq1</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> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">seq2</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>
<b>by </b><i><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">*</a> <font color="Maroon" title="c3">Ns</font></span>)</span> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">seq2</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>
<b>by </b><i><a class="ref" href="cfcont_1.html#T2" target="_self" title="CFCONT_1:th.2">Th2</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="E4:3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <font color="Maroon" title="c2">seq2</font></span>)</span> <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">*</a> <font color="Maroon" title="c3">Ns</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">*</a> <font color="Maroon" title="c3">Ns</font></span>)</span> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq2</font> <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">*</a> <font color="Maroon" title="c3">Ns</font></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"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="valued_1.html#K19" title="VALUED_1:func.19">(#)</a> <font color="Maroon" title="c2">seq2</font></span>)</span> <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">*</a> <font color="Maroon" title="c3">Ns</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">*</a> <font color="Maroon" title="c3">Ns</font></span>)</span> <a href="valued_1.html#K19" title="VALUED_1:func.19">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq2</font> <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">*</a> <font color="Maroon" title="c3">Ns</font></span>)</span></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_3"><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"><span class="p2">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="valued_1.html#K19" title="VALUED_1:func.19">(#)</a> <font color="Maroon" title="c2">seq2</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">*</a> <font color="Maroon" title="c3">Ns</font></span>)</span> <a href="valued_1.html#K19" title="VALUED_1:func.19">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">seq2</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><br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="valued_1.html#K19" title="VALUED_1:func.19">(#)</a> <font color="Maroon" title="c2">seq2</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 class="p1">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="valued_1.html#K19" title="VALUED_1:func.19">(#)</a> <font color="Maroon" title="c2">seq2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<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>
<b>by </b><i><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">seq1</font> <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> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq2</font> <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="valued_1.html#T5" title="VALUED_1:th.5">VALUED_1:5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">seq1</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> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq2</font> <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/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">seq1</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> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">seq2</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>
<b>by </b><i><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">*</a> <font color="Maroon" title="c3">Ns</font></span>)</span> <a href="valued_1.html#K19" title="VALUED_1:func.19">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">seq2</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>
<b>by </b><i><a class="ref" href="valued_1.html#T5" title="VALUED_1:th.5">VALUED_1:5</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="E6:3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="valued_1.html#K19" title="VALUED_1:func.19">(#)</a> <font color="Maroon" title="c2">seq2</font></span>)</span> <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">*</a> <font color="Maroon" title="c3">Ns</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">*</a> <font color="Maroon" title="c3">Ns</font></span>)</span> <a href="valued_1.html#K19" title="VALUED_1:func.19">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq2</font> <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">*</a> <font color="Maroon" title="c3">Ns</font></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>
