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<b>let </b><font color="Maroon" title="c1">s</font>, <font color="Maroon" title="c2">t</font> be   <a href="afinsq_1.html#NM1" title="AFINSQ_1:NM.1">XFinSequence</a>;<br/><b>let </b><font color="Maroon" title="c3">p</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/>





<i><font color="Green" title="E9">A1</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="c3">p</font> <a href="rewrite2.html#K3" title="REWRITE2:func.3">+^</a> <font color="Maroon" title="c2">t</font></span>)</span></span>)</span> = 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="rewrite2.html#K3" title="REWRITE2:func.3">+^</a> <font color="Maroon" title="c2">t</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 href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">p</font>
<b>by </b><i><a class="ref" href="rewrite2.html#D3" target="_self" title="REWRITE2:def.3">defConcatR</a></i>
<br/>.= 
 <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">p</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 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="c3">p</font></span>)</span> <a href="rewrite2.html#K3" title="REWRITE2:func.3">+^</a> <font color="Maroon" title="c2">t</font></span>)</span>
<b>by </b><i><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="18_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="E1:18_2"/><i><font color="Green" title="E10">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="c1">s</font> <a href="rewrite2.html#K2" title="REWRITE2:func.2">^+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="rewrite2.html#K3" title="REWRITE2:func.3">+^</a> <font color="Maroon" title="c2">t</font></span>)</span></span>)</span>
 ;<br/><a NAME="E2:18_2"/><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"><font color="Maroon" title="c3">p</font> <a href="rewrite2.html#K3" title="REWRITE2:func.3">+^</a> <font color="Maroon" title="c2">t</font></span>)</span>
 <b>by </b><i><a class="txt" href="rewrite2.html#E1:18_2"><i><font color="Green" title="E10">B2</font></i></a>, <a class="ref" href="rewrite2.html#D2" target="_self" title="REWRITE2:def.2">defConcatL</a></i>;<br/><a NAME="E3:18_2"/><i><font color="Green" title="E12">B</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="c3">p</font>
 <b>by </b><i><a class="txt" href="rewrite2.html#E2:18_2"><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/><a NAME="E4:18_2"/><i><font color="Green" title="E13">B3</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="c3">p</font></span>)</span>
 <b>by </b><i><a class="txt" href="rewrite2.html#E3:18_2"><i><font color="Green" title="E12">B</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="c3">p</font> <a href="rewrite2.html#K3" title="REWRITE2:func.3">+^</a> <font color="Maroon" title="c2">t</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="c3">p</font> <a href="rewrite2.html#K3" title="REWRITE2:func.3">+^</a> <font color="Maroon" title="c2">t</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#E2:18_2"><i><font color="Green" title="E11">B1</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"><span class="p2">(<span class="default"><font color="Maroon" title="c3">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="c2">t</font></span>)</span>
<b>by </b><i><a class="txt" href="rewrite2.html#E3:18_2"><i><font color="Green" title="E12">B</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="c1">s</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="rewrite2.html#K1" title="REWRITE2:func.1">.</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c2">t</font>
<b>by </b><i><a class="ref" href="afinsq_1.html#T30" title="AFINSQ_1:th.30">AFINSQ_1:30</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="c3">p</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="c2">t</font>
<b>by </b><i><a class="txt" href="rewrite2.html#E3:18_2"><i><font color="Green" title="E12">B</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="c3">p</font></span>)</span> <a href="rewrite2.html#K3" title="REWRITE2:func.3">+^</a> <font color="Maroon" title="c2">t</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#E4:18_2"><i><font color="Green" title="E13">B3</font></i></a>, <a class="ref" href="rewrite2.html#D3" target="_self" title="REWRITE2:def.3">defConcatR</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:18"/>
<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="c3">p</font> <a href="rewrite2.html#K3" title="REWRITE2:func.3">+^</a> <font color="Maroon" title="c2">t</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="c3">p</font></span>)</span> <a href="rewrite2.html#K3" title="REWRITE2:func.3">+^</a> <font color="Maroon" title="c2">t</font>
 <b>by </b><i><a class="txt" href="rewrite2.html#E1:18"><i><font color="Green" title="E9">A1</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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