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<div class="add">

<b>let </b><font color="Maroon" title="c1">s</font> be   <a href="afinsq_1.html#NM1" title="AFINSQ_1:NM.1">XFinSequence</a>;<br/><b>let </b><font color="Maroon" title="c2">p</font>, <font color="Maroon" title="c3">q</font> be  <a href="rewrite2.html#V1" title="REWRITE2:attr.1">XFinSequence-yielding</a> <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>;<br/>





 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="rewrite2.html#K2" title="REWRITE2:func.2">^+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c3">q</font></span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c3">q</font></span>)</span>
<b>by </b><i><a class="ref" href="rewrite2.html#T5" target="_self" title="REWRITE2:th.5">ThConcatLen</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">q</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="rewrite2.html#K2" title="REWRITE2:func.2">^+</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">q</font></span>)</span>
<b>by </b><i><a class="ref" href="rewrite2.html#T5" target="_self" title="REWRITE2:th.5">ThConcatLen</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="rewrite2.html#K2" title="REWRITE2:func.2">^+</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="rewrite2.html#K2" title="REWRITE2:func.2">^+</a> <font color="Maroon" title="c3">q</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rewrite2.html#T5" target="_self" title="REWRITE2:th.5">ThConcatLen</a></i>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="rewrite2.html#K2" title="REWRITE2:func.2">^+</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="rewrite2.html#K2" title="REWRITE2:func.2">^+</a> <font color="Maroon" title="c3">q</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>
;<br/>
<a NAME="E2:19"/><b>then </b><i><font color="Green" title="E9">A</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="rewrite2.html#K2" title="REWRITE2:func.2">^+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c3">q</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="rewrite2.html#K2" title="REWRITE2:func.2">^+</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="rewrite2.html#K2" title="REWRITE2:func.2">^+</a> <font color="Maroon" title="c3">q</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">k</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>assume </b><a NAME="E1:19_2"/><i><font color="Green" title="E10">B1</font></i>: 
<font color="Maroon" title="c4">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="rewrite2.html#K2" title="REWRITE2:func.2">^+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c3">q</font></span>)</span></span>)</span>
 ;<br/><a NAME="E2:19_2"/><i><font color="Green" title="E11">B2</font></i>: 
<font color="Maroon" title="c4">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c3">q</font></span>)</span>
 <b>by </b><i><a class="txt" href="rewrite2.html#E1:19_2"><i><font color="Green" title="E10">B1</font></i></a>, <a class="ref" href="rewrite2.html#D2" target="_self" title="REWRITE2:def.2">defConcatL</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:19_2_1_1"/>
( <font color="Maroon" title="c4">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">p</font> or  not <font color="Maroon" title="c4">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">p</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1_1"><b>suppose </b></a><a NAME="E1:19_2_1_1_1"/><i><font color="Green" title="E12">C1</font></i>: 
<font color="Maroon" title="c4">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">p</font>
 ;<br/><div class="add"><a NAME="E2:19_2_1_1_1"/><i><font color="Green" title="E13">C2</font></i>: 
<font color="Maroon" title="c4">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="rewrite2.html#K2" title="REWRITE2:func.2">^+</a> <font color="Maroon" title="c2">p</font></span>)</span>
 <b>by </b><i><a class="txt" href="rewrite2.html#E1:19_2_1_1_1"><i><font color="Green" title="E12">C1</font></i></a>, <a class="ref" href="rewrite2.html#D2" target="_self" title="REWRITE2:def.2">defConcatL</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="rewrite2.html#K2" title="REWRITE2:func.2">^+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c3">q</font></span>)</span></span>)</span> <a href="rewrite2.html#K1" title="REWRITE2:func.1">.</a> <font color="Maroon" title="c4">k</font> = 
<font color="Maroon" title="c1">s</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="rewrite2.html#K1" title="REWRITE2:func.1">.</a> <font color="Maroon" title="c4">k</font></span>)</span>
<b>by </b><i><a class="txt" href="rewrite2.html#E2:19_2"><i><font color="Green" title="E11">B2</font></i></a>, <a class="ref" href="rewrite2.html#D2" target="_self" title="REWRITE2:def.2">defConcatL</a></i>
<br/>.= 
<font color="Maroon" title="c1">s</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="rewrite2.html#K1" title="REWRITE2:func.1">.</a> <font color="Maroon" title="c4">k</font></span>)</span>
<b>by </b><i><a class="txt" href="rewrite2.html#E1:19_2_1_1_1"><i><font color="Green" title="E12">C1</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="rewrite2.html#K2" title="REWRITE2:func.2">^+</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="rewrite2.html#K1" title="REWRITE2:func.1">.</a> <font color="Maroon" title="c4">k</font>
<b>by </b><i><a class="txt" href="rewrite2.html#E1:19_2_1_1_1"><i><font color="Green" title="E12">C1</font></i></a>, <a class="ref" href="rewrite2.html#D2" target="_self" title="REWRITE2:def.2">defConcatL</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="rewrite2.html#K2" title="REWRITE2:func.2">^+</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="rewrite2.html#K2" title="REWRITE2:func.2">^+</a> <font color="Maroon" title="c3">q</font></span>)</span></span>)</span> <a href="rewrite2.html#K1" title="REWRITE2:func.1">.</a> <font color="Maroon" title="c4">k</font>
<b>by </b><i><a class="txt" href="rewrite2.html#E2:19_2_1_1_1"><i><font color="Green" title="E13">C2</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1_2"><b>suppose </b></a><a NAME="E1:19_2_1_1_2"/><i><font color="Green" title="E12">C'</font></i>: 
not <font color="Maroon" title="c4">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">p</font>
 ;<br/><div class="add"><a NAME="E2:19_2_1_1_2"/><i><font color="Green" title="E13">D1</font></i>: 
( <font color="Maroon" title="c4">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 or <font color="Maroon" title="c4">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">p</font> )
 <b>by </b><i><a class="txt" href="rewrite2.html#E1:19_2_1_1_2"><i><font color="Green" title="E12">C'</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:19_2_1_1_2"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">k</font> &amp; <font color="Maroon" title="c4">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c3">q</font></span>)</span> )
 <b>by </b><i><a class="txt" href="rewrite2.html#E2:19_2"><i><font color="Green" title="E11">B2</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>then consider </b><font color="Maroon" title="c5">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E5:19_2_1_1_2"/><i><font color="Green" title="E14">L</font></i>: 
( <font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">i</font> &amp; <font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">q</font> )
 <b>by </b><i><a class="txt" href="rewrite2.html#E2:19_2_1_1_2"><i><font color="Green" title="E13">D1</font></i></a>, <a class="ref" href="rewrite2.html#T2" target="_self" title="REWRITE2:th.2">ThSeq40</a></i>;<br/><a NAME="E6:19_2_1_1_2"/><i><font color="Green" title="E15">L1</font></i>: 
<font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">q</font>
 <b>by </b><i><a class="txt" href="rewrite2.html#E5:19_2_1_1_2"><i><font color="Green" title="E14">L</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E7:19_2_1_1_2"/><b>then </b><i><font color="Green" title="E16">L2</font></i>: 
<font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="rewrite2.html#K2" title="REWRITE2:func.2">^+</a> <font color="Maroon" title="c3">q</font></span>)</span>
 <b>by </b><i><a class="ref" href="rewrite2.html#D2" target="_self" title="REWRITE2:def.2">defConcatL</a></i>;<br/><a NAME="E8:19_2_1_1_2"/><i><font color="Green" title="E17">C3</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="rewrite2.html#K2" title="REWRITE2:func.2">^+</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">p</font>
 <b>by </b><i><a class="ref" href="rewrite2.html#T5" target="_self" title="REWRITE2:th.5">ThConcatLen</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="rewrite2.html#K2" title="REWRITE2:func.2">^+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c3">q</font></span>)</span></span>)</span> <a href="rewrite2.html#K1" title="REWRITE2:func.1">.</a> <font color="Maroon" title="c4">k</font> = 
<font color="Maroon" title="c1">s</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="rewrite2.html#K1" title="REWRITE2:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">i</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="rewrite2.html#E5:19_2_1_1_2"><i><font color="Green" title="E14">L</font></i></a>, <a class="txt" href="rewrite2.html#E2:19_2"><i><font color="Green" title="E11">B2</font></i></a>, <a class="ref" href="rewrite2.html#D2" target="_self" title="REWRITE2:def.2">defConcatL</a></i>
<br/>.= 
<font color="Maroon" title="c1">s</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="rewrite2.html#K1" title="REWRITE2:func.1">.</a> <font color="Maroon" title="c5">i</font></span>)</span>
<b>by </b><i><a class="txt" href="rewrite2.html#E6:19_2_1_1_2"><i><font color="Green" title="E15">L1</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="rewrite2.html#K2" title="REWRITE2:func.2">^+</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="rewrite2.html#K1" title="REWRITE2:func.1">.</a> <font color="Maroon" title="c5">i</font>
<b>by </b><i><a class="txt" href="rewrite2.html#E6:19_2_1_1_2"><i><font color="Green" title="E15">L1</font></i></a>, <a class="ref" href="rewrite2.html#D2" target="_self" title="REWRITE2:def.2">defConcatL</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="rewrite2.html#K2" title="REWRITE2:func.2">^+</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="rewrite2.html#K2" title="REWRITE2:func.2">^+</a> <font color="Maroon" title="c3">q</font></span>)</span></span>)</span> <a href="rewrite2.html#K1" title="REWRITE2:func.1">.</a> <font color="Maroon" title="c4">k</font>
<b>by </b><i><a class="txt" href="rewrite2.html#E5:19_2_1_1_2"><i><font color="Green" title="E14">L</font></i></a>, <a class="txt" href="rewrite2.html#E7:19_2_1_1_2"><i><font color="Green" title="E16">L2</font></i></a>, <a class="txt" href="rewrite2.html#E8:19_2_1_1_2"><i><font color="Green" title="E17">C3</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:19_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="rewrite2.html#K2" title="REWRITE2:func.2">^+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c3">q</font></span>)</span></span>)</span> <a href="rewrite2.html#K1" title="REWRITE2:func.1">.</a> <font color="Maroon" title="c4">k</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">s</font> <a href="rewrite2.html#K2" title="REWRITE2:func.2">^+</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="rewrite2.html#K2" title="REWRITE2:func.2">^+</a> <font color="Maroon" title="c3">q</font></span>)</span></span>)</span> <a href="rewrite2.html#K1" title="REWRITE2:func.1">.</a> <font color="Maroon" title="c4">k</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:19"/>
<font color="Maroon" title="c1">s</font> <a href="rewrite2.html#K2" title="REWRITE2:func.2">^+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="rewrite2.html#K2" title="REWRITE2:func.2">^+</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="rewrite2.html#K2" title="REWRITE2:func.2">^+</a> <font color="Maroon" title="c3">q</font></span>)</span>
 <b>by </b><i><a class="txt" href="rewrite2.html#E2:19"><i><font color="Green" title="E9">A</font></i></a>, <a class="ref" href="finseq_1.html#T17" title="FINSEQ_1:th.17">FINSEQ_1:17</a></i>;<br/>

 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="rewrite2.html#K3" title="REWRITE2:func.3">+^</a> <font color="Maroon" title="c1">s</font></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c3">q</font></span>)</span>
<b>by </b><i><a class="ref" href="rewrite2.html#T5" target="_self" title="REWRITE2:th.5">ThConcatLen</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">q</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="rewrite2.html#K3" title="REWRITE2:func.3">+^</a> <font color="Maroon" title="c1">s</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">q</font></span>)</span>
<b>by </b><i><a class="ref" href="rewrite2.html#T5" target="_self" title="REWRITE2:th.5">ThConcatLen</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="rewrite2.html#K3" title="REWRITE2:func.3">+^</a> <font color="Maroon" title="c1">s</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="rewrite2.html#K3" title="REWRITE2:func.3">+^</a> <font color="Maroon" title="c1">s</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rewrite2.html#T5" target="_self" title="REWRITE2:th.5">ThConcatLen</a></i>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="rewrite2.html#K3" title="REWRITE2:func.3">+^</a> <font color="Maroon" title="c1">s</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="rewrite2.html#K3" title="REWRITE2:func.3">+^</a> <font color="Maroon" title="c1">s</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>
;<br/>
<a NAME="E6:19"/><b>then </b><i><font color="Green" title="E10">A</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="rewrite2.html#K3" title="REWRITE2:func.3">+^</a> <font color="Maroon" title="c1">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="rewrite2.html#K3" title="REWRITE2:func.3">+^</a> <font color="Maroon" title="c1">s</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="rewrite2.html#K3" title="REWRITE2:func.3">+^</a> <font color="Maroon" title="c1">s</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">k</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>assume </b><a NAME="E1:19_4"/><i><font color="Green" title="E11">B1</font></i>: 
<font color="Maroon" title="c4">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="rewrite2.html#K3" title="REWRITE2:func.3">+^</a> <font color="Maroon" title="c1">s</font></span>)</span>
 ;<br/><a NAME="E2:19_4"/><i><font color="Green" title="E12">B2</font></i>: 
<font color="Maroon" title="c4">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c3">q</font></span>)</span>
 <b>by </b><i><a class="txt" href="rewrite2.html#E1:19_4"><i><font color="Green" title="E11">B1</font></i></a>, <a class="ref" href="rewrite2.html#D3" target="_self" title="REWRITE2:def.3">defConcatR</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_4_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:19_4_1_1"/>
( <font color="Maroon" title="c4">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">p</font> or  not <font color="Maroon" title="c4">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">p</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_4_1_1_1"><b>suppose </b></a><a NAME="E1:19_4_1_1_1"/><i><font color="Green" title="E13">C1</font></i>: 
<font color="Maroon" title="c4">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">p</font>
 ;<br/><div class="add"><a NAME="E2:19_4_1_1_1"/><i><font color="Green" title="E14">C2</font></i>: 
<font color="Maroon" title="c4">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="rewrite2.html#K3" title="REWRITE2:func.3">+^</a> <font color="Maroon" title="c1">s</font></span>)</span>
 <b>by </b><i><a class="txt" href="rewrite2.html#E1:19_4_1_1_1"><i><font color="Green" title="E13">C1</font></i></a>, <a class="ref" href="rewrite2.html#D3" target="_self" title="REWRITE2:def.3">defConcatR</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="rewrite2.html#K3" title="REWRITE2:func.3">+^</a> <font color="Maroon" title="c1">s</font></span>)</span> <a href="rewrite2.html#K1" title="REWRITE2:func.1">.</a> <font color="Maroon" title="c4">k</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="rewrite2.html#K1" title="REWRITE2:func.1">.</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c1">s</font>
<b>by </b><i><a class="txt" href="rewrite2.html#E2:19_4"><i><font color="Green" title="E12">B2</font></i></a>, <a class="ref" href="rewrite2.html#D3" target="_self" title="REWRITE2:def.3">defConcatR</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="rewrite2.html#K1" title="REWRITE2:func.1">.</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c1">s</font>
<b>by </b><i><a class="txt" href="rewrite2.html#E1:19_4_1_1_1"><i><font color="Green" title="E13">C1</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="rewrite2.html#K3" title="REWRITE2:func.3">+^</a> <font color="Maroon" title="c1">s</font></span>)</span> <a href="rewrite2.html#K1" title="REWRITE2:func.1">.</a> <font color="Maroon" title="c4">k</font>
<b>by </b><i><a class="txt" href="rewrite2.html#E1:19_4_1_1_1"><i><font color="Green" title="E13">C1</font></i></a>, <a class="ref" href="rewrite2.html#D3" target="_self" title="REWRITE2:def.3">defConcatR</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="rewrite2.html#K3" title="REWRITE2:func.3">+^</a> <font color="Maroon" title="c1">s</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="rewrite2.html#K3" title="REWRITE2:func.3">+^</a> <font color="Maroon" title="c1">s</font></span>)</span></span>)</span> <a href="rewrite2.html#K1" title="REWRITE2:func.1">.</a> <font color="Maroon" title="c4">k</font>
<b>by </b><i><a class="txt" href="rewrite2.html#E2:19_4_1_1_1"><i><font color="Green" title="E14">C2</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_4_1_1_2"><b>suppose </b></a><a NAME="E1:19_4_1_1_2"/><i><font color="Green" title="E13">C'</font></i>: 
not <font color="Maroon" title="c4">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">p</font>
 ;<br/><div class="add"><a NAME="E2:19_4_1_1_2"/><i><font color="Green" title="E14">D1</font></i>: 
( <font color="Maroon" title="c4">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 or <font color="Maroon" title="c4">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">p</font> )
 <b>by </b><i><a class="txt" href="rewrite2.html#E1:19_4_1_1_2"><i><font color="Green" title="E13">C'</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:19_4_1_1_2"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">k</font> &amp; <font color="Maroon" title="c4">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c3">q</font></span>)</span> )
 <b>by </b><i><a class="txt" href="rewrite2.html#E2:19_4"><i><font color="Green" title="E12">B2</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>then consider </b><font color="Maroon" title="c5">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E5:19_4_1_1_2"/><i><font color="Green" title="E15">L</font></i>: 
( <font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">i</font> &amp; <font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">q</font> )
 <b>by </b><i><a class="txt" href="rewrite2.html#E2:19_4_1_1_2"><i><font color="Green" title="E14">D1</font></i></a>, <a class="ref" href="rewrite2.html#T2" target="_self" title="REWRITE2:th.2">ThSeq40</a></i>;<br/><a NAME="E6:19_4_1_1_2"/><i><font color="Green" title="E16">L1</font></i>: 
<font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">q</font>
 <b>by </b><i><a class="txt" href="rewrite2.html#E5:19_4_1_1_2"><i><font color="Green" title="E15">L</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E7:19_4_1_1_2"/><b>then </b><i><font color="Green" title="E17">L2</font></i>: 
<font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="rewrite2.html#K3" title="REWRITE2:func.3">+^</a> <font color="Maroon" title="c1">s</font></span>)</span>
 <b>by </b><i><a class="ref" href="rewrite2.html#D3" target="_self" title="REWRITE2:def.3">defConcatR</a></i>;<br/><a NAME="E8:19_4_1_1_2"/><i><font color="Green" title="E18">C3</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="rewrite2.html#K3" title="REWRITE2:func.3">+^</a> <font color="Maroon" title="c1">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">p</font>
 <b>by </b><i><a class="ref" href="rewrite2.html#T5" target="_self" title="REWRITE2:th.5">ThConcatLen</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="rewrite2.html#K3" title="REWRITE2:func.3">+^</a> <font color="Maroon" title="c1">s</font></span>)</span> <a href="rewrite2.html#K1" title="REWRITE2:func.1">.</a> <font color="Maroon" title="c4">k</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="rewrite2.html#K1" title="REWRITE2:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">i</font></span>)</span></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c1">s</font>
<b>by </b><i><a class="txt" href="rewrite2.html#E5:19_4_1_1_2"><i><font color="Green" title="E15">L</font></i></a>, <a class="txt" href="rewrite2.html#E2:19_4"><i><font color="Green" title="E12">B2</font></i></a>, <a class="ref" href="rewrite2.html#D3" target="_self" title="REWRITE2:def.3">defConcatR</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="rewrite2.html#K1" title="REWRITE2:func.1">.</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c1">s</font>
<b>by </b><i><a class="txt" href="rewrite2.html#E6:19_4_1_1_2"><i><font color="Green" title="E16">L1</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="rewrite2.html#K3" title="REWRITE2:func.3">+^</a> <font color="Maroon" title="c1">s</font></span>)</span> <a href="rewrite2.html#K1" title="REWRITE2:func.1">.</a> <font color="Maroon" title="c5">i</font>
<b>by </b><i><a class="txt" href="rewrite2.html#E6:19_4_1_1_2"><i><font color="Green" title="E16">L1</font></i></a>, <a class="ref" href="rewrite2.html#D3" target="_self" title="REWRITE2:def.3">defConcatR</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="rewrite2.html#K3" title="REWRITE2:func.3">+^</a> <font color="Maroon" title="c1">s</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="rewrite2.html#K3" title="REWRITE2:func.3">+^</a> <font color="Maroon" title="c1">s</font></span>)</span></span>)</span> <a href="rewrite2.html#K1" title="REWRITE2:func.1">.</a> <font color="Maroon" title="c4">k</font>
<b>by </b><i><a class="txt" href="rewrite2.html#E5:19_4_1_1_2"><i><font color="Green" title="E15">L</font></i></a>, <a class="txt" href="rewrite2.html#E7:19_4_1_1_2"><i><font color="Green" title="E17">L2</font></i></a>, <a class="txt" href="rewrite2.html#E8:19_4_1_1_2"><i><font color="Green" title="E18">C3</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:19_4"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="rewrite2.html#K3" title="REWRITE2:func.3">+^</a> <font color="Maroon" title="c1">s</font></span>)</span> <a href="rewrite2.html#K1" title="REWRITE2:func.1">.</a> <font color="Maroon" title="c4">k</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="c2">p</font> <a href="rewrite2.html#K3" title="REWRITE2:func.3">+^</a> <font color="Maroon" title="c1">s</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="rewrite2.html#K3" title="REWRITE2:func.3">+^</a> <font color="Maroon" title="c1">s</font></span>)</span></span>)</span> <a href="rewrite2.html#K1" title="REWRITE2:func.1">.</a> <font color="Maroon" title="c4">k</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:19"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="rewrite2.html#K3" title="REWRITE2:func.3">+^</a> <font color="Maroon" title="c1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="rewrite2.html#K3" title="REWRITE2:func.3">+^</a> <font color="Maroon" title="c1">s</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="rewrite2.html#K3" title="REWRITE2:func.3">+^</a> <font color="Maroon" title="c1">s</font></span>)</span>
 <b>by </b><i><a class="txt" href="rewrite2.html#E6:19"><i><font color="Green" title="E10">A</font></i></a>, <a class="ref" href="finseq_1.html#T17" title="FINSEQ_1:th.17">FINSEQ_1:17</a></i>;<br/>


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